A symmetry-breaking bifurcation theorem and some related theorems applicable to maps having unbounded derivatives

We refine and generalize a symmetry-breaking bifurcation theorem by Werner and Spence [14]. Our theorem is so simple that we can apply it to the numerical verification for the bifurcation phenomena, for example, in non-linear vibration described by a semilinear wave equation. The point of our refinement is that the simplicity condition on (the candidate of) a bifurcation point in the original theorem is replaced by the regularity condition of a certain map, which is an easier condition to check. Our generalization enables us to apply the theorem directly to non-Frechet differentiable maps and makes the computational process simple. For the same purpose we also generalize some basic functional analytical theorems such as the convergence theorem of Newton's method and implicit function theorems.